Orbital Control of Single-Molecule Conductance Perturbed by Ï

Article pubs.acs.org/JPCC

Orbital Control of Single-Molecule Conductance Perturbed by π‑Accepting Anchor Groups: Cyanide and Isocyanide Junya Koga, Yuta Tsuji, and Kazunari Yoshizawa* Institute for Materials Chemistry and Engineering and International Research Center for Molecular Systems, Kyushu University, Fukuoka 819-0395, Japan S Supporting Information *

ABSTRACT: Electron transport properties through benzene molecules disubstituted with π-accepting cyanide and isocyanide anchor groups at their para and meta positions are investigated on the basis of a qualitative orbital analysis at the Hückel molecular orbital level of theory. The applicability of a previously derived orbital symmetry rule for electron transport is extended to the systems perturbed by the π-accepting anchor groups, where the HOMO−LUMO symmetry in the molecular orbital energies relative to the Fermi level is removed. The conservation of the HOMO−LUMO symmetry in the spatial distribution of the molecular orbitals between the unperturbed benzene molecule and the perturbed molecules with the anchor groups rationalizes symmetryallowed electron transport through the para isomers. On the other hand, destructive interferences between the nearly 2-fold degenerate frontier orbitals constructed from the 2-fold degenerate orbitals of the unperturbed benzene molecule and the anchor groups lead to symmetry-forbidden electron transport through the meta isomers. The qualitative orbital thinking is supported by more quantitative density functional theory (DFT) calculations combined with the nonequilibrium Green’s function (NEGF) method. The orbital analysis is a powerful tool for the understanding and rational design of molecular devices composed of πconjugated hydrocarbons and those perturbed by the π-accepting anchor groups. anchor groups, leading to fluctuation in conductance.33 Therefore anchor groups such as cyanide (−CN) and isocyanide (−NC) groups that have weaker coupling with gold electrodes have been receiving plenty of attentions since they can form more uniform contact geometries with a strong structural selectivity.20,21,34 In previous studies we have proposed an orbital symmetry rule for electron transport properties of single molecules from the analysis of Green’s function in terms of MOs.35−45 We have demonstrated that the rule is applicable to π−conjugated systems such as graphene sheets,35−37 small-size aromatic hydrocarbons,38,42,43 diarylethenes,39,40 polycyclic aromatic hydrocarbons,41 heterocyclic aromatic hydrocarbons,44 and cyclophanes.45 Not only site-specific electron transport phenomena but also the effect of torsional degrees of freedom on electron transport has been discussed by using the orbital symmetry rule.35−45 The rule is based on the nonequilibrium Green’s function (NEGF) method,46 which is commonly used to calculate coherent transport in metal−insulator−metal junctions. Theoretical studies based on Landauer’s formula47 with Green’s function techniques play an essential role in designing molecular electronic devices and detailed understanding of electron transport properties through a single molecule.48−54 The conductance of nanosized systems

1. INTRODUCTION As the miniaturization of microelectronic devices approaches their technological and physical limits, single-molecule electronic devices have gained great attention as an alternative to conventional semiconductor devices.1 Recent advances in nanofabrication techniques suggest a wide variety of potential applications of molecular devices such as switches,2 rectifiers,3 and memories.4 Mechanically controllable break junction (MCBJ)5 and scanning tunneling microscopy break junction (STM-BJ)6 techniques allow the measurement of electron transport down to the single-molecule level. In the break junction techniques, individual molecules are wired to two metal electrodes via proper anchor units. The effective coupling between molecular orbitals (MOs) and metal states is required for the realization of high conductance junction.7 Anchor units that play a crucial role in the coupling between a molecule and electrodes determine essential transport characteristics. Also, the energy level alignment between the frontier orbitals of a molecule and the Fermi level of metal electrodes is of great importance for transport characteristics.7 The position of frontier orbitals is influenced by anchor groups in accordance with their donating or accepting character.8 To date, a wide range of anchor units has been proposed, for example, thiol,9 amine,10 carboxylic acid,11 phosphine,12 phosphine sulfide,8 pyridine,6 cyanide,13−21 and isocyanide.14,16,22−32 Thiol is most often used to connect molecules with gold electrodes because of the high covalent bond strength. However, a large variability in its binding geometries is observed in systems with thiol © 2012 American Chemical Society

Received: July 10, 2012 Revised: August 27, 2012 Published: September 4, 2012 20607

dx.doi.org/10.1021/jp3068156 | J. Phys. Chem. C 2012, 116, 20607−20616

The Journal of Physical Chemistry C

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sufficiently large absolute value is required. The requirements for high transmission probability imposed on connecting sites to electrodes are summarized as follows: (1) Two atoms in which the sign of the product of the MO expansion coefficients in the HOMO is different from that in the LUMO should be connected with electrodes and (2) two atoms in which the orbital amplitudes of the HOMO and LUMO are significant should be connected with electrodes. For a degenerate system, where the energy of singly occupied molecular orbital 1 (SOMO1) is equal to that of SOMO2, the HOMO and LUMO in eq 8 are replaced by SOMO1 and SOMO2, as follows:42,43

consisting of a molecule and two electrodes in the limit of zero temperature and zero bias voltage is calculated from Landauer’s formula as follows:47 g=

2e 2 T (E F ) h

(1)

2

where 2e /h is the quantum conductance, T is the transmission probability, and EF is the Fermi level of the electrode. T is calculated with the following formula:46 T (E) = Tr[ΓR (E)GR (E)ΓL(E)G A (E)] R

(2)

A

where G and G are retarded and advanced Green’s functions for the metal−molecule−metal system, respectively; ΓL and ΓR are broadening functions for the left and right electrodes, respectively. The broadening function is defined as follows:46 A ΓL/R (E) = i[Σ RL/R − Σ L/R ]

Cr SOMO1Cs*SOMO1 C C* + r SOMO2 sSOMO2 E F − εSOMO1 ± iη E F − εSOMO2 ± iη

Since the denominators in eq 9 have the same sign, the opposite rule is applicable as a rule (1′): two atoms in which the sign of CrSOMO1CrSOMO1 * is the same as the sign of CrSOMO2CrSOMO2 * should be connected with electrodes. The necessary preconditions for the derivation of the orbital symmetry rule can be summarized as follows:35−37 (a) the coupling between the molecule and electrodes is weak, (b) there is electron−hole symmetry (pairing theorem) in orbital energies and MO expansion coefficients, and (c) the Fermi level is located at the midgap of the HOMO and LUMO. In previous studies we have tested the applicability of the rule to strong-coupled molecules with electrodes such as benzendithiol and naphthalenedithiol.42,43 Orbital interaction analysis using the fragment molecular orbital (FMO) method and density functional theory (DFT) calculations has indicated that the rule is valid in the Au−S bonding systems. The orbital phase on the anchoring sulfur atoms dominates the electron transport properties, which has been received experimental support.43 The HOMO and LUMO of the parent molecule are affected in a similar way by the 3p AO in the anchoring sulfur atoms since the energy level of the 3p AO in sulfur is very close to that of the 2p AO in carbon. Thus, the introduction of the sulfur anchor does not violate the preconditions (b) and (c). However, anchor groups with an electron-accepting character can decrease the LUMO level,8 which violates the preconditions (b) and (c). The orbital rule is useful for the development of a chemical way of thinking about electron transport in molecules in terms of frontier orbital theory.59 The purpose of this study is to test the applicability of the orbital symmetry rule to molecules with electron-accepting anchor groups such as cyanide (−CN) and isocyanide (-NC). Cyanide and isocyanide are well-known π-accepting ligands in coordination and organometallic chemistry. In recent years their ability to attach organic molecules to metal surfaces has attracted much attention. Although the strength of binding of isocyanide to gold surfaces is weaker than that of thiol, isocyanide groups can adsorb on gold surfaces, forming selfassembled monolayer (SAM).22,23,25,26,28 Experimental studies on the availability of isocyanide groups as an anchor unit have been widely reported,16,22,23,25,26,28,30−32 whereas only a limited number of experimental studies on cyanide groups have been carried out, producing conflicting results.16−21 Kiguchi et al. attempted conductance measurements of a cyanide-anchored molecule using the STM-BJ method but did not observe any clear conductance signatures.16 On the other hand, Mishchenko et al. performed a combined experimental and theoretical study of conductance through a cyanide-anchored molecule and reported well-defined peaks in the conductance histogram

(3)

where ∑L/R is the self-energy matrix for the left and right electrodes. The self-energy matrix is expressed as follows:46 ΣR/A (E) = τ †g R/A (E)τ

(4)

where τ is the molecule-electrode interaction, g (E) is the retarded and advanced Green’s functions of metal electrodes that do not interact with the molecule. We assume that the interaction between the metal electrode and the molecule is dominated by only the nearest neighbor atom’s interaction. Retarded and advanced Green’s functions are written as follows:55 R/A

GR/A (E) = [I − G(0)R/A (E)ΣR/A (E)]−1 G(0)R/A (E)

(5)

where I is the unit matrix and G(0)R/A is the zeroth Green’s function, which is the Green’s function of the isolated molecule. Equation 5 is derived from the following Dyson’s equation:55 G = G(0) + G(0)ΣG

(6)

If the interaction between the molecule and the electrode is weak, GR/A is considered to be proportional to G(0)R/A.56,57 The matrix elements of the zeroth Green’s function at the Fermi level, G(0)R/A (EF), which describes the electron transmission rs between atoms r and s, is written in terms of the MOs as follows:58 Grs(0)R/A (E F) =

∑ k

CrkCsk* E F − εk ± iη

(7)

where Crk is the kth MO cofficient at the rth atomic orbital (AO) in an orthogonal basis, εk is the kth MO energy, and η is an infinitesimal number. The expression of the zeroth Green’s function, eq 7, is useful to understand the relation between the MOs and electron transport. When we assume the Fermi level of the electrode to be located between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), the zeroth Green’s function is dominated by the contributions from the HOMO and LUMO, being approximated as follows: Cr HOMOCs*HOMO Cr LUMOCs*LUMO + E F − εHOMO ± iη E F − εLUMO ± iη

(9)

(8)

To obtain high conductance, these terms must be enhanced. Since the denominators of the respective terms have different signs, the reversal of the signs between the numerators with 20608



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resulting from the high selectivity of the N−Au binding.20 Many comparative studies of cyanide and isocyanide vs thiol have been performed experimentally and theoretically.13−16,19,21,23−25,28−32 However, depending on the experimental and theoretical approaches, the cyanide and isocyanide anchor groups have been found to show higher or lower conductance than thiol anchor groups, respectively. Detailed orbital analysis based on the phase, amplitude, and spatial distribution of the frontier orbitals can provide new insight into the electron transport properties through cyanide- and isocyanide-anchored systems. Site-specific electron transport phenomena through cyanide- and isocyanide-anchored systems are addressed in this manuscript on the basis of the qualitative orbital concept.

We performed electron transport calculations with NEGF method at the DFT level of theory (NEGF-DFT)48,49 to confirm the qualitative predictions obtained from the HMO calculations. We performed geometry optimizations of the isolated molecules with the Gaussian 09 program63 at the B3LYP/6-31G(d) level of theory.63−67 We used the ATK 11.2.3 program68 for quantitative electron transport calculations with a realistic two-probe model of molecular junction. The two-probe model used in this study consists of the semi-infinite left and right electrodes and central region. The left and right electrodes were modeled by two Au(111)-(3 × 3) surfaces (i.e., each layer includes 9 Au atoms). Three layers from each electrode (in total 54 Au atoms) were included in the central region. The local density approximation with the Perdew− Zunger parametrization (LDA-PZ) of the exchange-correlation functional was used.69 Owing to the coordinative nature of the covalent N−Au and C−Au bonds via the nitrogen and carbon lone pairs, the on-top binding site is suggested to be the most probable and stable adsorption site for cyanide and isocyanide70 anchor groups by experimental and theoretical studies.19,20,71−74 Optimizations of the central region geometry were carried out under a full open boundary condition, in which the cell vector along the electrode-electrode axis was optimized to determine the optimal N−Au and C−Au distances. To save computational efforts, the electrode atoms were frozen in their bulk geometries, the single-ζ basis set (SZ) was used for the gold atoms, and the double-ζ basis set with polarization (DZP) was used for all other atoms. Using the zero-bias optimized structures, we performed electron transport calculations that include the full self-consistent field (SCF) treatment, obtaining bias-dependent transmission spectra, current−voltage (I−V) curves, and molecular projected selfconsistent Hamiltonian (MPSH) states. The MPSH states are the eigenstates of the molecule within the two-probe environment and provide essential insight into the electron transport.49,68

2. COMPUTATIONAL METHODS We performed Hückel MO calculations to make qualitative predictions for the conductance of cyanide- and isocyanideanchored benzene molecules. Scheme 1 illustrates the geometry Scheme 1

of the molecules chosen for the study along with the shorthand nomenclature used throughout the remainder of this manuscript. A wide range of methods to approximate the Hamiltonian matrix elements for systems including heteroatoms within the Hückel framework is proposed.60 We employed αN = αC + 0.6βC−C as the Coulomb integral for nitrogen, βCN = 1.2βC−C as the resonance integral between nitrogen and carbon atoms in the functional groups, and βC−N = 0.9βC−C as that between isocyanide group and benzene. We determined these values after a careful comparative review of various studies in the literature.60 Using the formalism given in eqs 2−7, we combined the NEGF method with the Hückel molecular orbital method (NEGF-HMO) to obtain the transmission spectra and compared the results with the qualitative frontier orbital predictions. In addition to the α and β values above, we employed βAu−Au = 0.6βC−C as the resonance integral between gold atoms, βAu−N = 0.2βC−C as that between gold and nitrogen atoms, and βAu−C = 0.4βC−C as that between gold and carbon atoms to describe a one-dimensional gold chain and electrodemolecule coupling. We determined the value of βAu−Au based on the literature38 and calculated the values of βAu−C and βAu−N with the Wolfsberg−Helmholz approximation61 using the values of the overlap integral within the framework of the extended Hückel molecular orbital (eHMO) method.62 The overlap between the 5d orbital of gold atom and the 2p orbital of carbon and nitrogen atoms were taken into account.

3. RESULTS AND DISCUSSIONS We begin our study with qualitative thinking of the parent benzene molecule based on the HMO method. As shown in Figure 1, benzene possesses 2-fold degenerate e1g HOMOs and

Figure 1. Frontier orbitals of benzene and symmetry-allowed and symmetry-forbidden routes for electron transport.

e2u LUMOs. For connection 1−4 (para) the product of the MO coefficients on atoms 1 and 4 in the e11g HOMO is different in sign from that in the e12u LUMO. It is not necessary to consider the e21g HOMO and e22u LUMO since the amplitudes on atoms 1 and 4 are zero and these orbitals would not play a role in electron transport. According to the orbital symmetry rule, we expect that the para connection should have high conductance and it should be described as symmetry allowed for the transmission of electrons. For connection 2−6 (meta) two 20609



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Figure 2. MO energy diagrams near the Fermi level and frontier orbitals of (a) p-BDCN and (b) p-BDNC calculated with the HMO theory. The MOs of p-BDCN and p-BDNC are partitioned into the benzene molecule and anchor unit in accordance with the FMO method.

degenerated HOMOs and LUMOs should be taken into account. For the connection atom sites 2 and 6, electron−hole symmetry is observed between the e11g HOMO and e12u LUMO and between e21g HOMO and e22u LUMO. Since the product of the MO coefficients on atoms 2 and 6 in the e11g HOMO has the same sign as that in the e12u LUMO and the same holds true for the e21g HOMO and e22u LUMO, cancellation can occur between the e11g HOMO and e12u LUMO and the same holds true for the e21g HOMO and e22g LUMO. Thus, we expect that the meta connection should have low conductance and it should be described as symmetry forbidden. The MO energy diagrams and orbital distributions of pBDCN and p-BDNC calculated with the HMO method are shown in Figure 2, where the MOs are partitioned into the benzene molecule and anchor unit in accordance with the FMO method.75,76 Since the interaction between the two CN groups is negligible because of the large distance between them, the HOMOs and LUMOs of the anchor unit are almost degenerate. The essential features of the orbital interactions are identical between p-BDCN and p-BDNC. The LUMOs of the anchor unit have almost the same energy as those of the benzene molecule while the HOMOs of the anchor unit are lower than those of the benzene molecule, which leads to the

low-lying LUMO of cyanide- and isocyanide-anchored molecules. The b1 g HOMO of p-BDCN and p-BDNC is derived from the out-of-phase combination of the e11g HOMO of the benzene molecule and the πg HOMO of the anchor unit. The a2u LUMO of p-BDCN and p-BDNC is derived from the in-phase combination of the e12u LUMO of the benzene molecule and the πu LUMO of the anchor unit. In the πu LUMO of the anchor unit, cyanide has larger orbital amplitude on atoms interacting with the benzene molecule than isocyanide. Thus, the LUMO of the cyanide-anchored molecule is more stabilized than that of the isocyanide-anchored molecule, which implies that the cyanide groups have stronger electron-withdrawing properties than the isocyanide groups. Since the e21g HOMO and e22u LUMO of the benzene molecule have no orbital amplitude on the connecting sites, they have no interactions with the anchor unit. Thus, the degeneracy of the frontier orbitals in benzene is removed by the introduction of the anchor units. The combination of the gerade and ungerade orbitals in the frontier orbitals, which lead to the symmetryallowed connection, is conserved between benzene and the molecules with the anchor groups. Thus, p-BDCN and pBDNC are expected to be symmetry allowed as is the case with the para connection of benzene. Owing to the low-lying 20610



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Figure 3. MO energy diagrams near the Fermi level and frontier orbitals of (a) m-BDCN and (b) m-BDNC calculated with the HMO theory. The MOs of m-BDCN and m-BDNC are partitioned into the benzene molecule and anchor unit in accordance with the FMO method.

LUMO, electron−hole symmetry is removed in p-BDCN and p-BDNC, but it is not a significant problem since the constructive interference in eq 8 does not necessarily require the equivalent contributions from the HOMO and LUMO terms. The MO energy diagrams and orbital distributions of mBDCN and m-BDNC calculated with the HMO method are shown in Figure 3. The energy-level alignments between the frontier orbitals of the benzene molecule and those of the anchor unit in the meta-substituted molecules are very similar to those in the para-substituted molecules, but the way of orbital interaction is different between them. The essential features of the orbital interactions are identical between mBDCN and m-BDNC. The degenerate a2 and b2 HOMOs of the anchor unit and the degenerate e21g and e11g HOMOs of the benzene molecule are coupled out-of-phase to result in the nearly degenerate a2 HOMO and b2 HOMO-1 of m-BDCN and m-BDNC, respectively. The degenerate a2 and b2 LUMOs of the anchor unit and the degenerate e22u and e12u LUMOs of the benzene molecule are coupled in-phase to result in the nearly degenerate a2 LUMO and b2 LUMO+1 of m-BDCN and m-BDNC, respectively. A simple application of the orbital

symmetry rule between the HOMO and LUMO and between the HOMO-1 and LUMO+1 results in symmetry-forbidden connections because the signs of the product of the MO coefficients on the connecting sites are the same between them. However, for the case of symmetry-forbidden connections, the precondition of electron−hole symmetry is significant because the proper cancellation between the HOMO and LUMO terms in eq 8 requires these terms to have opposite sign and the same absolute value. The rule (1′), which is for a degenerate system, can avoid the problem of the broken electron−hole symmetry. The rule (1′) is based on eq 9, where the equality of the denominators between the two degenerate orbitals is axiomatic, and the proper cancellation between them requires the MO coefficients on the connecting sites to have opposite sign and the same amplitude. These requirements are satisfied in the nearly degenerate HOMO and HOMO-1 and in the LUMO and LUMO+1. Thus, m-BDCN and m-BDNC are expected to be symmetry forbidden as is the case with the meta connection of benzene. Computed transmission spectra for the cyanide- and isocyanide-anchored benzene molecules calculated with the NEGF-HMO method are shown in Figure 4. The essential 20611



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Figure 4. Transmission spectra near the Fermi level of (a) p- (red) and m-BDCN (blue) and (b) p- (red) and m-BDNC (blue) calculated with the NEGF-HMO method.

Figure 5. Computed transmission spectra for (a) p- (red) and m-BDCN (blue) and (b) p- (red) and m-BDNC (blue) at zero bias calculated with the NEGF-DFT method.

is two times larger than that of BDCN, has 1 order of magnitude larger transmission probability than BDCN. Although the para isomers have lower transmission probability than the meta isomers in the vicinity of the resonance peaks of the meta isomers, the para isomers have higher transmission probability than the meta isomers in most parts of the transmission spectra. Therefore the prediction based on the qualitative orbital analysis are in good agreement with the results of the NEGF-HMO calculations. To estimate the effects of the π-accepting anchor groups on the orbital symmetry rule, more quantitative calculations with the NEGF-DFT method were carried out by using the ATK program. Figure 5 shows obtained transmission spectra for BDCN and BDNC. Although the LUMO resonance peaks are much closer to the Fermi level (E = 0) than those observed in the NEGF-HMO results, the NEGF-DFT results denote almost the identical tendency of the NEGF-HMO results in the vicinity of the Fermi level. In the NEGF-DFT calculations the Fermi level was determined from DFT calculations of the bulk

features of the transmission spectra remain unchanged between BDCN and BDNC. The sharp transmission peaks come from the resonance tunneling effect at the location of MO levels. The larger peak broadening for BDNC results from a stronger coupling to the gold electrodes. As expected from the πaccepting nature of the cyanide and isocyanide anchor groups, the LUMO resonance peaks are closer to the Fermi level (E = 0) than the HOMO resonance peaks. The LUMO resonance peaks of BDCN is closer to the Fermi level than that of BDNC, which can be rationalized by the aforementioned qualitative orbital analysis that shows stronger electron-withdrawing properties of cyanide groups than isocyanide groups. The transmission probabilities at the Fermi level, which plays an important role in conductance, as mentioned earlier in eq 1, show a finite nonzero value for the symmetry-allowed para isomers, but those drop to zero for the symmetry-forbidden meta isomers. Since the transmission probability is proportional to the fourth power of the coupling strength between the gold electrode and anchor unit,56,57 BDNC, whose coupling strength 20612



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Figure 6. MPSH states of (a) p-BDCN, (b) p-BDNC, (c) m-BDCN, and (d) m-BDNC near the Fermi level.

gold electrodes. On the other hand, in the NEGF-HMO calculations the Fermi level is assumed to be located at the origin of the energy (E = 0) since the electrodes are described by an ideal semi-infinite one-dimensional chain with the same on-site energy as that of the 2p AO in carbon. Although the definition of the Fermi level will affect the general profiles of

the transmission spectra, we think that this simple definition is a good starting point for qualitative discussion based on this simple model. In the NEGF-DFT results we can observe the shift of the position of interference features.77 The electron transport through BDCN and BDNC takes place via the tail of the nearby LUMO levels. For p-BDCN and p-BDNC, which 20613



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given, any physical quantity is directly calculated without solving the secular equation, by making the analysis of the determinant.83 However, expressions of physical quantities with MO coefficients are commonly used. The orbital symmetry rule is based on the expression of the Green’s function with MO coefficients. The chemical understanding of physical properties with MO coefficients is in line with the frontier orbital theory84 and the Woodward−Hoffmann rules85 for chemical reactions. According to the Landauer−Büttiker formula, the current is obtained after integration of a finite part of the transmission spectra called bias window as follows:46

are expected as symmetry-allowed connections, the transmission probabilities at the Fermi level are 0.12 and 0.16, respectively. On the other hand, for m-BDCN and m-BDNC, which are expected as symmetry-forbidden connections, the transmission probabilities at the Fermi level are 0.013 and 0.006, respectively. Only π-tunneling is considered in the NEGF-HMO calculations, while both π- and σ-tunneling mechanisms are taken into account in the NEGF-DFT calculations. Thus, the striking difference between symmetryallowed and -forbidden connections observed in the NEGFHMO results, such as transmission dips in the symmetryforbidden connections, is weakened in the NEGF-DFT results. However, the NEGF-DFT results are consistent with the qualitative prediction based on the orbital analysis with the HMO method. We show in Figure 6 MPSH states and spatial distributions of the four π-type orbitals closest to the Fermi level for the para and meta isomers. The MPSH states, the eigenstates of the molecule within the two-probe environment, do not include the self-energies of the electrodes. The complex portion of the selfenergy broadens the transmission peaks, while the real part will give a shift of the transmission peaks relative to the MPSH states. Therefore the transmission peaks and the MPSH energies do not always coincide. In these figures the MPSH states composed of the σ-type orbitals, states 25 and 26, are not shown and orbitals extended to the electrodes are excluded for clarity. Although the BDNC molecules have a lower LUMO than the BDCN molecules, due to the stronger coupling of isocyanide groups to the electrodes than cyanide groups, the energy level alignments and orbital distributions of the MPSH states are almost all consistent with the qualitative HMO calculations. DFT calculations on the isolated molecules show that the BDCN molecules have a lower LUMO than the BDNC molecules. For the para isomers, state 23, which corresponds to the LUMO in the isolated molecule, plays a crucial role in electron transport. This orbital analysis is fully consistent with thermopower measurements, in which the cyanide groups change transport relative to benzene dithiol such that transport is dominated by the LUMO in 1,4BDCN.17 The second nearest state to the Fermi level, state 24, does not contribute to electron transport since there is no orbital amplitude on atoms connected to electrodes. On the other hand, for the meta isomers, both states 23 and 24 play a crucial role in electron transport. Thus, two transmission peaks are observed in the vicinity of the Fermi level. Since the signs of the products of the MO coefficients on the connecting sites vary between these orbitals, destructive contributions from these orbitals at the Fermi level are expected, which agrees well with the qualitative orbital analysis based on the HMO method as well as quantum interference discussion.78−82 The orbital symmetry rule is closely related to the quantum interference discussion since the both methods are based on the Green’s function formalism. Recently Markussen et al. demonstrated that quantum interference in aromatic molecules is intimately related to the topology of the molecules’s π system and established a simple graphical scheme to predict the existence of quantum-interference-induced transmission antiresonances.82 In their method, the Green’s function formalism is reduced to the Hückel determinant and conditions for the antiresonances are derived from a direct relationship between the Hückel determinant and the topology of the molecules’s π system without use of the orbital analysis. As pointed out by Coulson and Longuet-Higgins, once the Hückel determinant is

I (V ) =

2e h

eV /2

∫−eV /2 dET(E , V )

(10)

The bias window includes only energies close to the Fermi level in the interval from −eV/2 to eV/2, which corresponds to the electrochemical potentials of the left and right electrodes.46,48 The peaks close to the Fermi level have crucial effects on the low-bias current. Figure 7 shows I−V curves for BDCN and

Figure 7. Computed I-V curves of BDCN and BDNC with the NEGFDFT method.

BDNC computed with the NEGF-DFT method. For both BDCN and BDNC, the para isomers have larger current than the meta isomers. These NEGF-DFT results are in good agreement with the qualitative predictions using the HMO method. The current of p-BDCN, m-BDCN, and p-BDNC is saturated at higher voltages, where the whole transmission peak is included in the bias window. The current of m-BDNC is not saturated, due to the broad transmission peak of m-BDNC lying apart from the Fermi level. The I−V calculation provides a clear view of the electron transport behavior, demonstrating the applicability of the orbital symmetry rule to cyanide- and isocyanide-anchored systems.

4. SUMMARY AND CONCLUSIONS We have investigated effects of cyanide and isocyanide πaccepting anchor groups on the orbital symmetry rule for the electron transport properties through a π-conjugated molecule. We employed benzene disubstituted with the anchor groups at their para and meta positions as a model system (i.e., p-BDCN, 20614



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p-BDNC, m-BDCN, and m-BDNC). According to the orbital control rule based on the qualitative analysis within the HMO framework, in the parent benzene molecule, the electron transport in the para direction is symmetry-allowed while that in the meta direction is symmetry-forbidden. The cyanide and isocyanide anchor groups with strong electron-withdrawing nature yield molecules with low-lying LUMO levels, leading to a breakdown of the HOMO−LUMO symmetry in the MO energies relative to the Fermi level. For p-BDCN and p-BDNC, however, the conservation of the HOMO−LUMO symmetry in the spatial distribution of the MOs between benzene and the anchored molecules (i.e., the combination of the gerade HOMO and ungerade LUMO), which is clearly demonstrated by the qualitative FMO analysis, rationalizes the symmetryallowed electron transport through p-BDCN and p-BDNC. For m-BDCN and m-BDNC, nearly 2-fold degenerate frontier orbitals are formed from the out-of-phase and in-phase combinations of the 2-fold degenerate frontier orbitals of the fragments. According to the modified orbital symmetry rule, which can avoid the precondition of the HOMO−LUMO symmetry in the MO energies relative to the Fermi level, destructive contributions from these nearly 2-fold degenerate frontier orbitals at the Fermi level result in the symmetryforbidden electron transport through m-BDCN and m-BDNC. We performed calculations of the transmission spectra with the NEGF-HMO method, obtaining fully consistent results with the qualitative orbital views. We also compared the qualitative orbital analysis and the NEGF-HMO results with calculations performed with the NEGF-DFT method using realistic molecular junction models. The transmission spectra, orbital distributions of the MPSH states, and I−V curves calculated with the NEGF-DFT method verified the qualitative expectations for the electron transport through the perturbed molecules. The applicability of the orbital symmetry rule is successfully extended to molecules with π-accepting anchor groups that significantly alter the HOMO and LUMO levels relative to the Fermi level.

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ASSOCIATED CONTENT

S Supporting Information *

Atomic Cartesian coordinates for the optimized geometries of the central regions with p-BDCN, p-BDNC, m-BDCN, and mBDNC and complete ref 63. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS K.Y. is thankful for Grants-in-Aid for Scientific Research (Nos. 22245028 and 24109014) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), the Kyushu University Global COE Project, the Nanotechnology Support Project, the MEXT Project of Integrated Research on Chemical Synthesis, and CREST of the Japan Science and Technology Cooperation. Y.T. thanks JSPS for a graduate fellowship. 20615



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